key term - Vertical Compression

Definition

Vertical compression is a transformation that involves scaling a function vertically, effectively shrinking or expanding the function along the y-axis. This transformation alters the amplitude or range of the function while preserving its general shape and behavior.

5 Must Know Facts For Your Next Test

1. Vertical compression of a function $f(x)$ is achieved by multiplying the function by a constant $a$, where $0 < |a| < 1$, resulting in the new function $f(x) = a \cdot f(x)$.
2. Vertical expansion of a function $f(x)$ is achieved by multiplying the function by a constant $a$, where $|a| > 1$, resulting in the new function $f(x) = a \cdot f(x)$.
3. Vertical compression and expansion affect the amplitude or range of the function, but do not change the function's period or frequency.
4. Vertical compression and expansion can be used to model real-world phenomena, such as the relationship between volume and pressure in Boyle's law.
5. Vertical compression and expansion are important transformations in the study of basic classes of functions, as they allow for the manipulation and analysis of function behavior.

Review Questions

• Explain how vertical compression affects the amplitude or range of a function.
  • Vertical compression of a function $f(x)$ involves multiplying the function by a constant $a$, where $0 < |a| < 1$. This has the effect of shrinking the function along the y-axis, effectively reducing the amplitude or range of the function. For example, if the original function $f(x)$ has a maximum value of 10 and a minimum value of -10, then vertically compressing it by a factor of $a = 0.5$ would result in a new function $f(x) = 0.5 \cdot f(x)$ with a maximum value of 5 and a minimum value of -5, while preserving the general shape and behavior of the function.
• Describe how vertical compression and expansion can be used to model real-world phenomena.
  • Vertical compression and expansion are useful transformations for modeling various real-world relationships and phenomena. For example, in Boyle's law, which describes the inverse relationship between the pressure and volume of a gas, the function can be represented as $V = k/P$, where $V$ is the volume, $P$ is the pressure, and $k$ is a constant. By vertically compressing or expanding this function, we can model how changes in pressure affect the volume of the gas, which is important in fields like engineering and physics. Similarly, vertical transformations can be used to model the relationship between voltage and current in electrical circuits, or the relationship between force and acceleration in classical mechanics.
• Analyze how vertical compression and expansion, along with other function transformations, can be used to study the behavior of basic classes of functions.
  • Vertical compression and expansion, along with other function transformations such as horizontal shifts, reflections, and combinations of these, are essential tools for studying the behavior of basic classes of functions, including linear, quadratic, polynomial, exponential, and trigonometric functions. By applying these transformations, we can manipulate the functions to explore their properties, such as domain, range, symmetry, and periodicity. This allows us to gain a deeper understanding of the underlying characteristics of these function classes and how they can be used to model and analyze real-world phenomena. The ability to transform functions vertically, horizontally, and through other means is a fundamental skill in the study of basic classes of functions and their applications.

Related terms

Dilation: A transformation that scales a function either vertically or horizontally, resulting in the function being stretched or compressed.

Amplitude:

The maximum displacement of a periodic function from its midline or average value.

Reflection:

A transformation that flips a function across a line, either the x-axis or y-axis, resulting in a mirrored image of the original function.